So today I went to a local range and rented a Beretta 92FS, I've only fired this handgun a few times before, and this was the longest I've ever fired any handgun in double action.

All the first shots were made in double action. I did notice, that as I continued firing and fatigue set in, I started having more issues maintaining my groups and keeping them centered on my target. After 50 rounds through the handgun, I was landing nearly all my shots right of center (in the "Thumbing" section of the target).

I tried a number of different things, such as relaxing my right hands thumb a little, the old push/pull method of shooting, ensuring I'm not squeezing the handgun with my entire hand/fingers among other things. I was also starting to shoot high towards the end of my shooting session, where it says I'm "Breaking Wrist Up" which I'm also unsure what exactly that means.

20 shots were made from roughly 7yrd/21ft, off hand in 5 round intervals. Ammo was PMC Bronze 9mm 115grn. 5 shots were shot at 25yrds (Was screwing around at that point, hence the flyers off to the sides) 25 shots total

That's not at all bad shooting, although 50 rds shouldn't be enough to tire you out.
Were all the shots done one or two handed?
Were the shots slow and deliberate or fast?
If two handed, you might not have been using the support hand well.
Most of the strength of the grip comes from the support hand and the wrists.
If done one handed, the stance might not have been aggressive enough.
Breaking the wrists is a common problem and is just what it sounds like.
Wrist(s) should be locked straight, not bent.

Diagnostic targets do provide some -- limited! -- value. For someone shooting slow fire, and shooting one handed from a traditional bullseye stance. They work best for someone who already really understands the fundamentals and may just need a reminder as to details.

But they aren't really useful for people shooting two handed.

Or for people shooting rapidly instead of slow fire.

They don't help people who might have multiple, overlapping issues to solve.

And they aren't particularly valuable for people who don't already have the fundamentals nailed down, cold.

****

All that said? "Thumbing" happens when you're shooting with your thumb up or forward, relaxed, and then you tense it up and squeeze it down or into the gun as you press the trigger.

Another cause of bullets landing in that same area is getting too much finger on the trigger, so that as you press the trigger you actually "hook" the trigger around, pulling the muzzle off target to the right. (Not having seen you shoot, my money would be on this one, with a DA trigger on an unfamiliar gun. If your hands are large, you may be trying to get a little extra leverage by pressing the side of the trigger instead of its center.)

Another cause of bullets landing in that area: if you're pushing against the side of the gun with your left thumb while shooting two-handed, as people sometimes do when they're first getting used to a new pistol.

All of that can be compounded with a heel push or breaking the wrist up (which are basically the same action). That's a not-uncommon flinch pattern some people do when they're new to the gun.

There's also some vision possibilities and a few other things, but those are the major ones.

All shots were made with a two handed hold, in a slow methodical manner (2-3 seconds between each shot). I had to stop on several occasions to calm my hands down, I noticed that after 25 rounds, whenever I tried to shoot in double action, about half way through the trigger pull, my sights would begin to shake badly as I struggled with the weight of the trigger pull. Single action did not pose the same issue.

I tried adjusting where my finger rested on the trigger, and applying equal pressure, though I suspect that I may not have been doing it correctly.

Stance was essentially what I was told to do in basic, feet roughly shoulder width apart, with one foot forward and the other back with the knees slightly bent, leaning into the gun to provide as much support as possible.

The two handed grip is very hard to explain long distance.
Most new shooters think they are doing it correctly, but they have way too much grip with their shooting hand, and not nearly enough with the support hand, (the left one for a right handed shooter).
And that can cause the shaking and fatigue you are experiencing.
When the gun goes off, does it pop out of the support hand?
How limber is your trigger finger?
With your two handed grip, is it flexible or stiff?
The better the support hand is being used correctly, the easier it is to pull the trigger straight back.
Try using the thumbs forward technique, both thumbs pointed at the target, and push the gun forward with both hands and wrists, even getting the shoulder muscles involved.
It's worth a try to help with overcoming the shakes, the trigger pull effort and pointing the gun straight.
It's a technique that older shooters use to compensate for creaky elbow and wrist joints.

@g.willikers My support hand never separated when I fired from what I remember, but I do remember my right hand getting tired much more (Not sure if it's simply because I was squeezing the entire gun while pulling the trigger, recoil fatigue, wasn't used to the DA weight or all of the above.

I tried the push/pull method before, that's something my uncle had taught me back when I started shooting with him, where you pull with your support hand and push with your firing hand in an effort to gain a more stable shooting stance or something along those lines.

I typically try to lock my right wrist while firing and I don't "cup" the gun, instead wrapping my left hand over my right with the thumbs overlapping each other and parallel to the slide of the gun.

As pointed out, it's hard to really address your issues on-line, though I hope we're pointing you in the right direction...my two cents:
Start doing grip-strengthening exercises; it sounds as if your hand is simply getting tired. Shooters often fail to realize how much exercise can improve shooting, by improving strength and endurance.

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For sure, hand and wrist exercises can really help.
But use a soft rubber or foam ball to squeeze, rather than the spring type exercisers.
They give a more useful motion and won't take skin off if you slip.

Buy one of the afore mentioned soft foam or rubber balls.
I use one shaped like a little football, because it's shaped more like a pistol grip than a round one.
But either shape works.
Extend your arm, one hand at a time, like you would hold and aim a pistol, and squeeze and hold as long as you can.
Then repeat as much as you can.
This will strengthen hands, wrists and forearms, without harm.
Curling exercises with weights works, too, but using weights requires good instruction to avoid possible damage to wrists, elbows and shoulders.

From the picture you provided, the grouping is not good enough to give a definitive recommendation. As you can see, the rounds landed in several areas. That makes it hard to figure out what is happening. If you use the same style target and land a solid majority in the same section (in a nice group) I'd feel more confident in calling your error.

Right now, I'd say, go slow, take breaks as necessary, then get back to us.

Next time I'm out at the range, I'll see if I can get some video or something showing what I'm doing with my hands. I'll try the strength exercises as well and see where that takes me. There are a few certafied NRA trainers that oversee a local range, perhaps I can talk to them as well and see what they think as well.

May look into some handgun classes as well sometime late, but that's for a different thread and for a different day.

Thank you all for the feedback, hopefully I'll be able to do something that will help my groups.

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